A JACOBI-TYPE FORMULA FOR A FAMILY OF HYPERELLIPTIC CURVES OF GENUS 3 WITH AUTOMORPHISM OF ORDER 4

In this paper we show a two-dimensional variant of the classical Jacobi formula between a theta constant and the Gauss hypergeometric function. We use the family of algebraic curves given in the form w^4 = z^2(z − 1)^2(z − λ_1)(z − λ_2) with two complex parameters λ_1, λ_2 and the modular functions for them. Our result is an exact extension of the classical formula that is contained as a degenerated case. As an application we give a new proof for the extended Gauss arithmetic geometric mean theorem in two variables obtained by Koike and Shiga (J. Number Theory 128 (2008), 2029–2126)